Metatheory

Is there a cubical type theory? that describes more exactly what happens in the model of type theory in cubical sets??

Show in HoTT that the nthn^{th} universe is a model of HoTT with (n−1)(n-1) universes.

Consider the type theory with a sequence of universes plus for each n>0n\gt 0, an axiom asserting that there are n-types? that are not (n−1)(n-1)-types. Does adding axioms asserting that each universe is univalent increase the logical strength?

Homotopy theory

Prove that the torus? (with the HIT definition involving a 2-dimensional path) is equivalent to a product of two circles. (Has Kristina done this?)